Chapter 2 – Molecular Orbital Theory
Big-picture:
Now that we understand aspects of molecular structure, we can look in more detail at
bonding – how atoms are bonded together, energetics of bonding, where electrons reside in
bonds, etc. These characteristics of a molecule allow us to explain and predict even more of
its properties and applications.
Learning goals:
• Be able to construct molecular orbital diagrams for homonuclear diatomic,
heteronuclear diatomic, homonuclear triatomic, and heteronuclear triatomic molecules.
• Understand and be able to articulate how molecular orbitals form – conceptually,
visually, graphically, and (semi)mathematically.
• Interrelate bond order, bond length, and bond strength for diatomic and triatomic
molecules, including neutral and ionized forms.
• Use molecular orbital theory to predict molecular geometry for simple triatomic systems
• Rationalize molecular structure for several specific systems in terms of orbital overlap
and bonding.
Textbook Chapter 2
O2
Draw the Lewis dot structure for O2.
Compare to the experimentally observed chemistry of O2.
What does the Lewis dot structure get right, and what does it get wrong?
O2
Introduction to MO theory
Valence bond theory does a reasonable job of explaining the bonding in
some simple molecules, and therefore it is an appropriate first
approximation.
However, it fails to accurately describe important features of even simple
molecules such as O2, and (as you can imagine) it does not do a good job
with even more complex polyatomic molecules.
Parallel to what we have experienced with the development of models that
describe the structure of an atom, simple theories are a great starting
point, but more sophisticated models are needed to more accurately
account for all observations.
Molecular orbital (MO) theory is a more sophisticated quantum mechanical
model of bonding in molecules that can be applied successfully to both
simple and complex molecules.
Introduction to MO theory
Atomic orbitals are regions of space in which electrons have a high
probability of residing – electrons are “spread out” over the orbitals
that “surround” (or “comprise”) an atom.
Molecular orbitals can be thought of as a natural extension of this
concept: they are orbitals that are spread over all of the atoms in a
molecule and therefore they describe where electrons reside within
the molecule.
We will consider MO theory fairly qualitatively, but as with atomic
orbitals, it can be treated much more quantitatively.
High-level calculations using MO theory can be quite accurate, doing
a good job of predicting bond lengths, vibration frequencies,
vibrational spectra, electronic spectra, NMR spectra, …
Brief review of atomic orbitals (AOs)
De Broglie, 1924: Wave-particle duality describes electromagnetic radiation
and matter (such as electrons, protons...)
Schrodinger equation: Accounts for wave-particle duality, the motion of
electrons in an atom, and quantized nature of atomic structure
Total energy =
Eψ
=
KE
KEψ
+
+
PE
PEψ
Wavefunctions (ψ) and Atomic Orbitals
The Schrodinger equation is a 2nd order differential
equation that can be solved exactly for hydrogen, but
only numerically for many-electron atoms and
molecules. Only certain solutions produce physically
acceptable results – quantization.
Wavefunction: Solution to the Schrodinger equation,
describes the behavior of an electron moving in x, y,
and z directions – where it is and what it is doing
Each wavefunction that is a solution to the Schrodinger
equation is an atomic orbital
Quantum numbers uniquely label each orbital
Probability density
The probability of finding an electron at a particular point in space is
proportional to the square of the wavefunction at that point (x, y, z)
vs.
Nodes
Nodes: Regions where wavefunctions pass through zero
1s
2s
3s
Radial nodes: Radial component of the wavefunction passes through zero
Angular nodes: Angular component of the wavefunction passes through zero
Counting nodes
There are always n-1 nodes, where n is the principal quantum number
The number of angular nodes is ℓ
• Which orbitals are these?
• How many radial and angular nodes does each one have?
• How many radial and angular nodes does a 5d orbital have?
Making MOs from AOs
As with atomic orbitals, Schrodinger equations can be written for electrons
in molecules. Approximate solutions to these equations can be constructed
from linear combinations of atomic orbitals (LCAO), which are the sums and
differences of the atomic wave functions.
The coefficients can be equal or unequal (when?), positive or negative
(when?), depending on the individual orbitals and their energies.
Making MOs from AOs
As two atoms approach and the distance between them decreases, their
orbitals overlap, and in the overlap region there is a significant probability
for electrons to be found – why (qualitatively and quantitatively)?
Molecular orbitals form from this overlap, and electrons in bonding
molecular orbitals have a high probability of occupying the space between
nuclei – the electrostatic forces between the electrons and the two positive
nuclei hold the atoms together.
LCAO’s: Bonding for H 1s
LCAO’s: Antibonding for H 1s
What do these MOs look like?
How are their energies related?
MOs from p-orbital overlap
How can the p-orbitals overlap, and
what are their shapes and relative energies?
MOs from p-orbital overlap
MO diagram for O2
What all does this predict?
Considerations
Three conditions are essential for overlap to lead to bonding:
Symmetries
Energies
Distance
Considerations
What is the relationship between the number of atomic orbitals and the
number of molecular orbitals?
Bond order, bond length, bond strength
How are these correlated? Use O2 and its ions as an example.
Orbital mixing
The energies of the molecular orbitals depend on the degree of overlap,
which depends on symmetries and the energies of the constituent atomic
orbitals.
So far we considered only interactions of orbitals that have identical
energies. However, orbitals with similar, but not equal, energies can also
interact if they have appropriate symmetries.
Are there any orbitals you would predict to overlap in this way?
Based on symmetry,
which can interact?
Based on energy,
which can interact?
2p
2s
MO diagram for N2
N2, O2, F2, Ne2?
Heteronuclear diatomic molecules
How do heteronuclear MOs differ from homonuclear MOs?
(qualitatively and quantitatively)
How do we predict relative orbital energies?
This can be approximated by electronegativity, which tracks with relative
orbital energies (bonding electrons held close to atom).
HCl
What would you predict the MO
diagram for HCl to look like?
(Account for all orbitals)
Bonding continuum
Progression from covalent to ionic bonding shown nicely with MO theory
(not a hard, definitive cut-off)
HF
What would you predict the MO diagram for HF to look like?
(Account for all orbitals)
How does this compare to the Lewis dot structure you would draw?
What additional information does it provide?
Polyatomic molecules
Recall: N atomic orbitals yields N molecular orbitals.
We are not limited to diatomic molecules.
Consider, for example, a simple triatomic.
We can treat this much more quantitatively, integrating symmetry
considerations, but we will approach this more intuitively and qualitatively.
The FHF– ion is linear, with hydrogen in the middle.
Construct the MO diagram for FHF–
How do we begin?
FHF–
As with HF, only H-1s and F-2s,2p are energetically similar enough to interact.
Begin by putting H-1s (spherically symmetric) in the middle and looking
at how it overlaps – symmetry, phase, energy – with all permutations
of the F-2s and F-2p orbitals on each side.
FHF–
How can H-1s interact with F-2s? How significant will this interaction be?
FHF–
How can H-1s interact with F-2pz? How significant will this interaction be?
FHF–
How can H-1s interact with F-2px,y? How significant will this interaction be?
FHF– molecular orbital diagram
Using MO theory to predict molecular geometry
How can we predict the shape of a molecule using MO theory?
Consider the H3 molecule. Pretend it is linear.
What H-1s orbital configurations are possible?
H3+ molecular orbital diagram
H3: From linear to bent
What happens if we begin with H3 linear, and then bend it?
H3: From linear to bent
What happens if we begin with H3 linear, and then bend it?
H3: From linear to bent
H3: From linear to bent
Walsh diagram
H3+ vs H3–
What does this suggest about the molecular geometry of H3+ vs. H3–?
π-bonding: 2nd row vs. 3rd (4th, 5th, 6th) rows
Ethylene: Stable molecule, doesn't polymerize
without a catalyst.
Silylene: Never isolated, spontaneously polymerizes.
The large Ne core of Si atoms inhibits sideways overlap of
3p orbitals → weak π-bond
N can make π-bonds, so N2 has a very strong triple
bond and is a relatively inert diatomic gas
“RTV” silicone polymer (4 single bonds to Si)
vs. acetone (C=O double bond)
p-d orbital interactions
Other orbitals that are similar in energy and symmetry can also interact
d-d orbital interactions